Question

5. Find the solution set of the inequality 230 = 10c + 30.

Answer 1

5) Find the solution set of the inequality $230 \leq 10c + 30$

From given,

$230 \leq 10c + 30$

$\mathrm{Switch\:sides}\\\\10c+30\ge \:230\\\\\mathrm{Subtract\:}30\mathrm{\:from\:both\:sides}\\\\10c+30-30\ge \:230-30\\\\\mathrm{Simplify}\\\\10c\ge \:200\\\\\mathrm{Divide\:both\:sides\:by\:}10\\\\\frac{10c}{10}\ge \frac{200}{10}\\\\\mathrm{Simplify}\\\\c\ge \:20$

The solution set is:

$230\le \:10c+30\quad :\quad \begin{bmatrix}\mathrm{Solution:}\:&\:c\ge \:20\:\\ \:\mathrm{Interval\:Notation:}&\:[20,\:\infty \:)\end{bmatrix}$

6)

Write an inequality that represents all solutions to $4(2x + 8) \leq 40$

From given,

$4(2x + 8) \leq 40$

$\mathrm{Divide\:both\:sides\:by\:}4$

$\frac{4\left(2x+8\right)}{4}\le \frac{40}{4}$

$\mathrm{Simplify}\\\\2x+8\le \:10\\\\\mathrm{Subtract\:}8\mathrm{\:from\:both\:sides}\\\\2x+8-8\le \:10-8\\\\\mathrm{Simplify}\\2x\le \:2\\\\\mathrm{Divide\:both\:sides\:by\:}2\\\\\frac{2x}{2}\le \frac{2}{2}\\\\\mathrm{Simplify}\\\\x\le \:1$

The solution set is:

$4\left(2x+8\right)\le \:40\quad :\quad \begin{bmatrix}\mathrm{Solution:}\:&\:x\le \:1\:\\ \:\mathrm{Interval\:Notation:}&\:(-\infty \:,\:1]\end{bmatrix}$

6)

What is the solution set of the inequality 6(2 - r) < 18 ?

From given,

$6(2 - r) < 18$

$\mathrm{Divide\:both\:sides\:by\:}6\\\\\frac{6\left(2-r\right)}{6}$

When, we multiply or divide both sides by negative number, we must flip the inequality sign

$\left(-r\right)\left(-1\right)>1\cdot \left(-1\right)\\\\\mathrm{Simplify}\\\\r>-1$

The solution set is:

$6\left(2-r\right)-1\:\\ \:\mathrm{Interval\:Notation:}&\:\left(-1,\:\infty \:\right)\end{bmatrix}$

7)

Solve the inequality 4(-3b + 4) - b > -16b + 12

From given,

$4(-3b + 4) - b > -16b + 12$

$-12b + 16 - b > -16b + 12\\\\-13b + 16>-16b+12\\\\\mathrm{Subtract\:}16\mathrm{\:from\:both\:sides}\\\\-13b+16-16>-16b+12-16\\\\\mathrm{Simplify}\\\\-13b>-16b-4\\\\\mathrm{Add\:}16b\mathrm{\:to\:both\:sides}\\\\-13b+16b>-16b-4+16b\\\\\mathrm{Simplify}\\\\3b>-4\\\\\mathrm{Divide\:both\:sides\:by\:}3\\\\\frac{3b}{3}>\frac{-4}{3}\\\\\mathrm{Simplify}\\\\b>-\frac{4}{3}$

The solution set is:

$\left(-3b+4\right)-b>-16b+12\quad :\quad \begin{bmatrix}\mathrm{Solution:}\:&\:b>-\frac{4}{3}\\\\\:\mathrm{Interval\:Notation:}&\:\left(-\frac{4}{3},\:\infty \:\right)\end{bmatrix}$